We study the disturbance decoupling problem for linear time invariant descriptor systems. We give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability via a proportional and/or derivative feedback that also makes the resulting closed-loop system regular and/or of index at most one. All results are proved constructively based on condensed forms that can be computed using orthogonal matrix transformations, i.e., transformations that can be implemented in a numerically stable way.